# Difference between revisions of "ApCoCoA-1:Higman groups"

From ApCoCoAWiki

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elements a,b,c and d with the following relations: | elements a,b,c and d with the following relations: | ||

H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}> | H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}> | ||

− | + | ||

+ | ==== Reference ==== | ||

+ | Graham Higman, "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series 26 (1): 61–64, 1951. | ||

==== Computation ==== | ==== Computation ==== |

## Revision as of 10:47, 23 August 2013

#### Description

The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the elements a,b,c and d with the following relations:

H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}>

#### Reference

Graham Higman, "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series 26 (1): 61–64, 1951.

#### Computation

/*Use the ApCoCoA package ncpoly.*/ // a is invers to e, b is invers to f, c is invers to g and d is invers to h Use ZZ/(2)[a,b,c,d,e,f,g,h]; NC.SetOrdering("LLEX"); Define CreateRelationsHigman() Relations:=[]; // add the relation of the invers element Append(Relations,[[a,e],[1]]); Append(Relations,[[e,a],[1]]); Append(Relations,[[b,f],[1]]); Append(Relations,[[f,b],[1]]); Append(Relations,[[c,g],[1]]); Append(Relations,[[g,c],[1]]); Append(Relations,[[d,h],[1]]); Append(Relations,[[h,d],[1]]); // add the relation a^{-1}ba = b^2 Append(Relations,[[e,b,a],[b^2]]); // add the relation b^{-1}cb = c^2 Append(Relations,[[f,c,b],[c^2]]); // add the relation c^{-1}dc = d^2 Append(Relations, [[g,d,c],[d^2]]); // add the relation d^{-1}ad = a^2 Append(Relations, [[h,a,d],[a^2]]); Return Relations; EndDefine; Relations:=CreateRelationsHigman(); Relations; GB:=NC.GB(Relations,31,1,100,1000); GB; Size(GB);